Topic outline

  • Outcomes

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    In this course you will learn how to:

    1. Use exponential notation and exponential identities.
    2. Multiply powers.
    3. Divide powers.
    4. Raise a power to a power.
    5. Change fractional exponents to radical form.
    6. Simplify surds.

  • Exponential notation

    What are exponents?

    Exponents are a powerful way to mathematically describe rapid increases or decreases in growth. Exponential notation is very useful to describe very large and very small numbers.

    Just as multiplication is a short way to write repeated addition, similarly, exponents are a short way to write repeated multiplication. We discuss this further in the example that follows the video. 

    The power of exponential growth can be seen using a chessboard and grains of rice watch the next video to see this in action.

    • Example

      Look at the following expressions. Which do you think would be easier to write and work with? \(2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\) or \(2^{10}\)?
      Certainly \({{2}^{10}}\) is quicker to write and leaves less room for error.

      But how did we get from \(2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\) to
      \(2^{10}\)?

      Count the number of times that \(2\) has been multiplied by itself. What do you get?

      You would have counted that \(2\) is multiplied \(10\) times. The number ‘floating above’ the \(2\), which is called the exponent, is also \(10\). The exponent tells us how many times the base \(2\) has been multiplied by itself.

      \(2^{10}\) is called exponential form or a power. The base is the number that is repeatedly multiplied. The exponent which may also be referred to as the ‘index’ or ‘power’ tells us how many times the base is multiplied by itself.

      Exponential form explained

      We read \(2^{10}\) as \(2\) to the power of \(10\). This tells us that the base \(2\) has been multiplied by itself \(10\) times

      Think you've got it? Then try the activity on writing in exponential form.

    • Quiz icon
      Activity: Write in exponential form Quiz

    • Exponential identities

      There is an important identity that we use often in exponents. The identity states that any base raised to the power of zero is equal to one or \({{a}^{0}}=1\). 

      So far in the examples you have mainly used positive exponents but exponents can be negative too. Negative exponents produce fractions. You will use the following identity when dealing with negative exponents: \({a}^{-m}=\frac{1}{{a}^{+m}}\).

      The table below will help you understand how to change from negative to positive exponents.

      Converting negative exponents
      NEGATIVE EXPONENT    AS POSITIVE EXPONENT       SIMPLIFIED
      \({{3}^{-2}}\) \({{3}^{-2}}=\frac{1}{{{3}^{+2}}}\) \({{3}^{-2}}=\frac{1}{{{3}^{+2}}}=\frac{1}{9}\)
      \({{(2x)}^{-2}}\) \({{(2x)}^{-2}}=\frac{1}{{{(2x)}^{2}}}\) \({{(2x)}^{-2}}=\frac{1}{{{(2x)}^{2}}}=\frac{1}{4{{x}^{2}}}\)
      \(\frac{1}{{{2}^{-3}}}\) \(\frac{1}{{{2}^{-3}}}={{2}^{+3}}\) \(\frac{1}{{{2}^{-3}}}={{2}^{+3}}=8\)


      Click on the link below to take you to the next activity to check your understanding of negative exponents.

      Negative exponents practise

  • Multiply powers

    When we multiply powers with the same base, we keep the base and add the exponents. This can be generalised as:

    \({{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\text{ }\)

    To see examples using the law for multiplying powers with the same base, watch the next video.

  • Divide powers

    When we divide powers with the same base, we keep the base and subtract the exponents. This can be generalised as:

    \[\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\text{ }\]

    You can learn more about dividing powers with the same base by watching the next video.

  • Raise a power to a power

    When we raise a power to a power we multiply the exponents. This is generalised as the following exponent law:

     \[ { (a^m) }^{ n} = a^{(m\times n)} \] 

    This is called the exponent power rule.

    Raising a power to a power is explained in more detail in the next video. After you have watched all 3 videos on the laws of exponents try the interactive "exponent rules" activity to test your knowledge.

  • Radical form

    Exponents can be any rational number, which means they can be fractions too. A fractional exponent is the same as finding some root of a number. We generalise this as:
    \[{ x}^{\frac{1}{n} }=\sqrt[x]{n}\] where \(n\in N\) and \(x\in R\) 
    The root symbol has a special name, it is called a radical.

    Each part of a radical has its own name.

    Each part of a radical
    The next video shows examples of fractional exponents.

    • Change from fractional to radical form

      The number in the numerator of the fractional exponent becomes the exponent of the radicand and the number in the denominator of the fractional exponent is the root as shown in the next image. Convert from fractional exponent to radical form

      You can practise changing from rational exponents to radical form by clicking: Rational exponents exercise.

  • Simplify surds

    Some radicals can be written as rational numbers but some radicals cannot be written as rational numbers and we can only work out a rough estimate of their value. So it is best to leave them in radical form for example, \(\sqrt {55}\). We call these irrational roots, surds. In other words, surds are roots that cannot be reduced to a whole number or fraction. 

    Leaving a root in surd form is easier and more accurate than writing and rounding off the decimal value. However, there are methods we can use to simplify surds. 

    The methods to simplify surds are shown in the next video. Complete the quiz after you are certain you understand the rules for surds.

    What are surds?